The Restricted Edge-Connectivity of de Bruijn Undirected Graphs

A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity λ′ of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each component has at least two vertices. It has been shown by A. H. Esfahanian and S. L. Hakimi [On computing a conditional edge-connectivity of a graph. Information Processing Letters, 27(1988), 195-199] that λ′(G) ≤ ξ(G) for any graph of order at least four that is not a star, where ξ(G) = min{dG(u) + dG(v) − 2 : uv is an edge in G}. A graph G is called λ′-optimal if λ′(G) = ξ(G). This paper proves that the de Bruijn undirected graph UB(d, n) is λ′-optimal except UB(2, 1), UB(3, 1) and UB(2, 3) and, hence, is super edge-connected for n ≥ 1 and d ≥ 2.